It is well known that vari­ous stat­ist­ics of a large sample (of size \( n \)) are ap­prox­im­ately dis­trib­uted ac­cord­ing to the nor­mal law. The asymp­tot­ic ex­pan­sion of the dis­tri­bu­tion of the stat­ist­ic in a series of powers of \( n^{-1/2} \) with a re­mainder term gives the ac­cur­acy of the ap­prox­im­a­tion. H. Cramér [1937] first ob­tained the asymp­tot­ic ex­pan­sion of the mean, and re­cently P. L. Hsu [1945] has ob­tained that of the vari­ance of a sample. In the present pa­per we ex­tend the Cramér–Hsu meth­od to Stu­dent’s stat­ist­ic. The the­or­em proved states es­sen­tially that if the pop­u­la­tion dis­tri­bu­tion is non-sin­gu­lar and if the ex­ist­ence of a suf­fi­cient num­ber of mo­ments is as­sumed, then an asymp­tot­ic ex­pan­sion can be ob­tained with the ap­pro­pri­ate re­mainder. The first four terms of the ex­pan­sion are ex­hib­ited in for­mula (35).

Let \( X_1, X_2, \dots, X_n, \dots \) be in­de­pend­ent ran­dom vari­ables and let \( S_n = \sum_{\nu=1}^n X_{\nu} \). In the so-called law of the it­er­ated log­ar­ithm, com­pletely solved by Feller re­cently, the up­per lim­it of \( S_n \) as \( n\to\infty \) is con­sidered and its true or­der of mag­nitude is found with prob­ab­il­ity one. A coun­ter­part to that prob­lem is to con­sider the lower lim­it of \( S_n \) as \( n\to\infty \) and to make a state­ment about its or­der of mag­nitude with prob­ab­il­ity one.

The lim­it­ing dis­tri­bu­tion of the max­im­um cu­mu­lat­ive sum of a se­quence of in­de­pend­ent ran­dom vari­ables has been dis­cussed re­cently by Er­dős–Kac [1946] and Wald [1947]. Er­dős and Kac treated the case where each ran­dom vari­ables has zero mean, while Wald con­sidered more gen­er­al cases. We shall show that the prob­lem can be treated by a uni­form meth­od start­ing with a clas­sic­al com­bin­at­or­i­al for­mula due to De Moivre.

In the clas­sic­al coint-toss­ing game we have a se­quence of in­de­pend­ent ran­dom vari­ables \( X_{\nu} \), \( \nu=1,2,\dots \), each tak­ing the val­ues \( \pm 1 \) with prob­ab­il­ity \( 1/2 \). We are in­ter­ested in the signs of the par­tial sums \( S_n = \sum_{\nu=1}^n X_{\nu} \).

We con­sider a se­quence of in­de­pend­ent ran­dom vari­ables hav­ing the com­mon dis­tri­bu­tion func­tion \( F(x) \) which is as­sumed to be con­tinu­ous. Let \( nF_n(x) \) de­note the num­ber of ran­dom vari­ables among the first \( n \) of the se­quence whose val­ues do not ex­ceed \( x \). Write
\[ d_n = \sup_{-\infty < x < \infty}|n(F_n(x)-F(x))| .\]
Kolmogoroff [1933] proved that the prob­ab­il­ity
\begin{equation*}\tag{1}
P(d_n\leq\lambda n^{1/2}),
\end{equation*}
where \( \lambda \) is a pos­it­ive con­stant, tends as \( n\to\infty \) uni­formly in \( \lambda \) to the lim­it­ing dis­tri­bu­tion
\begin{equation*}\tag{2}
\Phi(\lambda)=\sum_{-\infty}^{\infty}(-1)^j e^{-2j^2\lambda^2}.
\end{equation*}
Smirnoff [1939] ex­ten­ded this res­ult and re­cently [Feller 1948] has giv­en new proofs of these the­or­ems. In this pa­per we shall ob­tain an es­tim­ate of the dif­fer­ence between (1) and (2) as a func­tion of \( n \), val­id not only for \( \lambda \) equal to a con­stant but also for \( \lambda \) equal to a func­tion \( \lambda(n) \) of \( n \) which does not grow too fast.

One as­pect of the the­ory of ad­di­tion of in­de­pend­ent ran­dom vari­ables is the fre­quency with which the par­tial sums change sign. In­vest­ig­a­tions of this nature were ori­gin­ated by Paul Lévy, in a pa­per [1939] which con­tains a wealth of ideas. This prob­lem as such was men­tioned by Feller in his 1945 ad­dress. In the case where the par­tial sums can ac­tu­ally van­ish the prob­lem falls un­der the head of “re­cur­rent events,” a gen­er­al the­ory of which was re­cently de­veloped in a pa­per by Feller [1949]. A very spe­cial case had been stud­ied in de­tail by Hunt and my­self [Chung and Hunt 1949]. Gen­er­al­iz­ing the prob­lem in a nat­ur­al way we shall con­sider the num­ber of times \( T_n \) with which the se­quence of re­duced par­tial sums \( S_k-E(S_k) \), \( k=1,2,\dots, n \) crosses a giv­en value \( c \). We shall es­tab­lish the lim­it­ing dis­tri­bu­tion of \( T_n \) in the case where the ran­dom
vari­ables have a com­mon dis­trib­utino with a fi­nite third ab­so­lute mo­ment.

Re­new­al the­ory has been treated by many pure and ap­plied math­em­aticians. Among the former we may men­tion Feller, Täck­lind and Doob. The prin­cip­al lim­it the­or­em (for one-di­men­sion­al, pos­it­ive, lat­tice ran­dom vari­ables) was however proved earli­er by Kolmogorov in 1936 as the er­god­ic the­or­em for de­nu­mer­able Markov chains. A par­tial res­ult for the non-lat­tice case was first proved by Doob us­ing the the­ory of Markov pro­cesses, and the com­plete res­ult by Black­well. The ex­ten­sion of the re­new­al the­or­em to ran­dom vari­ables tak­ing both pos­it­ive and neg­at­ive val­ues was first giv­en by Wolfow­itz and the au­thor [Chung and Wolfow­itz 1952], for the lat­tice case. A par­tial res­ult for the non-lat­tice case, us­ing a purely ana­lyt­ic­al ap­proach, was ob­tained by Pol­lard and the au­thor [Chung and Pol­lard 1952]. For the lit­er­at­ure see [Chung and Wolfow­itz 1952].

The fun­da­ment­als of the the­ory of de­nu­mer­able Markov chains with sta­tion­ary trans­ition prob­ab­il­it­ies were laid down by Kolmogorov and fur­ther work was done by Dob­lin. The the­ory of re­cur­rent events of Feller is closely re­lated, if not co­ex­tens­ive. Some new res­ults ob­tained by T. E. Har­ris turn out to tie up very nicely with some amp­li­fic­a­tions of Dob­lin’s work. Har­ris was led to con­sider the prob­ab­il­it­ies of hit­ting one state be­fore an­oth­er, start­ing from a third one. This idea of con­sid­er­ing three states, one ini­tial, one “ta­boo”, and one fi­nal, is more fully de­veloped in the present work. The no­tion of first pas­sage time of the “uni­on” or “in­ter­sec­tion” of two states is also in­tro­duced here. The in­ter­play between these no­tions is il­lus­trated.

Asymp­tot­ic prop­er­ties are es­tab­lished for the Rob­bins–Monro [1951] pro­ced­ure of stochastic­ally solv­ing the equa­tion \( M(x) = \alpha \). Two dis­joint cases are treated in de­tail. The first may be called the “bounded” case, in which the as­sump­tions we make are sim­il­ar to those in the second case of Rob­bins and Monro. The second may be called the “quasi-lin­ear” case which re­stricts \( M(x) \) to lie between two straight lines with fi­nite and non­van­ish­ing slopes but pos­tu­lates only the bounded­ness of the mo­ments of \( Y(x) - M(x) \). In both cases it is shown how to choose the se­quence \( \{a_n\} \) in or­der to es­tab­lish the cor­rect or­der of mag­nitude of the mo­ments of \( x_n - \theta \). Asymp­tot­ic nor­mal­ity of \( a^{1/2}_n(x_n - \theta) \) is proved in both cases un­der a fur­ther as­sump­tion. The case of a lin­ear \( M(x) \) is dis­cussed to point up oth­er pos­sib­il­it­ies. The stat­ist­ic­al sig­ni­fic­ance of our res­ults is sketched.

This pa­per is ex­clus­ively con­cerned with con­tinu­ous para­met­er Markov pro­cesses with a de­nu­mer­ably in­fin­ite num­ber of states and sta­tion­ary trans­ition mat­rix func­tion. The found­a­tions of the prop­er the­ory of such pro­cesses, as dis­tin­guished from that of the dis­crete para­met­er ver­sion, or of Markov pro­cesses which are either more spe­cial (for ex­ample, gen­er­al state space; non­station­ary trans­ition mat­rix func­tion), were laid by Doob [1942; 1945], and Lévy [1951; 1952; 1953]. Roughly speak­ing it was Lévy in 1952 who drew, in his in­im­it­able way, the com­pre­hens­ive pic­ture while Doob, ten years earli­er, had sup­plied the es­sen­tial in­gredi­ents. The present ef­for aims at a syn­thes­is of the most fun­da­ment­al parts of the the­ory, made pos­sible by the con­tri­bu­tions of these two au­thors. While the res­ults giv­en here gen­er­ally ex­tend and cla­ri­fy those in the cited lit­er­at­ure, im­mense cred­it must go to Pro­fess­ors Lévy and Doob for the in­spir­a­tion of their pi­on­eer work. To them I am also in­debted for much valu­able dis­cus­sion through cor­res­pond­ence and con­ver­sa­tion. An at­tempt is made in the present­a­tion to be quite form­al and rig­or­ous, in the spir­it of Doob’s already clas­sic treat­ise [1953]. Fur­ther de­vel­op­ments of the the­ory will be pub­lished else­where.

The pur­pose of this pa­per is to point out some new con­nec­tions between the sample func­tion be­ha­vi­or and the ana­lyt­ic­al prop­er­ties of the trans­ition prob­ab­il­ity func­tions of a con­tinu­ous para­met­er Markov chain with sta­tion­ary trans­ition prob­ab­il­it­ies. The main idea is that of a post-exit pro­cess de­rived from the ori­gin­al pro­cess by con­sid­er­ing its evol­u­tion after the exit from a stable state. This leads to vari­ous re­la­tions between con­di­tion­al prob­ab­il­it­ies all of which are “in­tu­it­ively ob­vi­ous” but re­quire some­times painstak­ing proofs if prob­ab­il­ity the­ory like any oth­er branch of math­em­at­ics is to be treated as a dis­cip­line in lo­gic. These re­la­tions im­ply cer­tain ana­lyt­ic­al prop­er­ties of the trans­ition func­tions, ob­tained re­cently by D. G. Aus­tin by purely ana­lyt­ic­al means, and ex­hib­it them in con­nec­tion with oth­er quant­it­ies in­tro­duced in this pa­per. They also com­plete some res­ults due to Doob and Lévy. The well-known dif­fer­en­tial equa­tions of Kolmogorov are seen to be lim­it­ing cases of cer­tain more gen­er­ally val­id dif­fer­en­tial equa­tions in­volving a con­tinu­ous para­met­er. One of these ex­presses the fun­da­ment­al trans­ition prop­erty of the post-exit pro­cess, and the oth­er a sim­il­ar prop­erty of the re­new­al dens­ity of an im­bed­ded re­new­al pro­cess.

Let \( \{X_i\},\ i=1,2,\dots \) be a se­quence of in­de­pend­ent and identic­ally dis­trib­uted in­teg­ral val­ued ran­dom vari­ables such that 1 is the ab­so­lute value of the greatest com­mon di­visor of all val­ues of \( x \) for which \( P(X_i=x) > 0 \). Define \( S_n=\sum_{i=1}^n X_i \). Chung and Fuchs [1951] showed that if \( x \) is any in­teger, \( S_n=x \) in­fin­itely of­ten with prob­ab­il­ity 1 ac­cord­ing as \( EX_i=0 \) or \( \neq 0 \), provided that \( E|X_i| < \infty \). Let \( 0 < EX_i < \infty \), and \( A \) de­note a set of in­tegers con­tain­ing an in­fin­ite num­ber of pos­it­ive in­tegers. It will be shown that any such set \( A \) will be vis­ited in­fin­itely of­ten with prob­ab­il­ity 1 by the se­quence \( \{S_n\}\ n=1,2,\dots \). Con­di­tions are giv­en so that sim­il­ar res­ults hold for the case where \( X_i \) has a con­tinu­ous dis­tri­bu­tion and the set \( A \) is a Le­besgue meas­ur­able set whose in­ter­sec­tion with the pos­it­ivfe real num­bers has in­fin­ite Le­besgue meas­ure.

The main ques­tion dis­cussed in this pa­per has been in the air for some time. Roughly speak­ing it may be stated as fol­lows: Giv­en a con­tinu­ous para­met­er Markov pro­cess \( \{x_t,\ t\geq 0\} \) with sta­tion­ary trans­ition prob­ab­il­it­ies, let us start it off at a ran­dom time \( \alpha \) chosen ac­cord­ing to the way it has been un­rav­el­ling, as it were, but without pre­vi­sion of the fu­ture. Does the shif­ted pro­cess \( \{x_{\alpha+t},\ t\geq 0\} \) re­main Markovi­an with the same trans­ition prob­ab­il­it­ies, and, know­ing \( x_{\alpha} \), is the past \( \{x_t,\ t\leq\alpha\} \) in­de­pend­ent of the fu­ture \( \{x_t,\ t\geq\alpha\} \)?

We give a very short proof of the re­cur­rence the­or­em of Chung and Fuchs [1951] in one and two di­men­sions. This new ele­ment­ary proof does not de­tract from the old one which uses a sys­tem­at­ic meth­od based on the char­ac­ter­ist­ic func­tion and yields a sat­is­fact­ory gen­er­al cri­terion. But the present meth­od, be­sides its brev­ity, also throws light on the com­bin­at­or­i­al struc­ture of the prob­lem.

The pur­pose of this pa­per is to present a simple uni­fied ap­proach to a group of the­or­ems in semi-group the­ory called the “ex­po­nen­tial for­mu­las”, due to Hille, Phil­lips, Wid­der and D. G. Kend­all [Hille and Phil­lips 1957, p. 354]. A more gen­er­al and ap­par­ently new for­mula is ar­rivefd at, which in­cludes some known cases. It turns out that these for­mu­las are in es­sence sum­mab­il­ity meth­ods which are best com­pre­hen­ded from the point of view of ele­ment­ary prob­ab­il­ity the­ory. They are all in the spir­it of S. Bern­stein’s proof of Wei­er­strass’s ap­prox­im­a­tion the­or­em, the same idea be­ing present in M. Riesz’s proof of Hille’s first ex­po­nen­tial for­mula (see [Hille et al. 1957, p. 314]). Where­as the de­tails here are just a little sim­pler than in [Hille and Phil­lips 1957], it seems of some in­terest to ex­hib­it the gen­er­al pat­tern and to re­duce the proofs to routine veri­fic­a­tions. The read­er who is not ac­quain­ted with the lan­guage of prob­ab­il­ity should have no dif­fi­culty in everything in­to the lan­guage of clas­sic­al ana­lys­is. But the prob­ab­il­ity way of think­ing is really ger­mane to the sub­ject.

The Mar­tin bound­ary for a dis­crete para­met­er Markov chain was first con­sidered by Doob, us­ing the the­ory of R. S. Mar­tin after whom the bound­ary is named. A dir­ect and in­geni­ous meth­od was later found by Hunt, who also strengthened Doob’s res­ults in sev­er­al points. In this note, I shall sketch a nat­ur­al ap­proach to the the­ory in the con­tinu­ous para­met­er case.

This pa­per may be re­garded as a new and fairly self-con­tained one at­tached to §§2–4 of [Chung 1963]. These sec­tions are en­titled “Ter­min­o­logy and nota­tion”, “The bound­ary” and “Fun­da­ment­al the­or­ems” re­spect­ively. The rest of [Chung 1963] is either con­tained in a more gen­er­al treat­ment (§5, §9 and parts of §6), or may be set aside as spe­cial cases un­der ad­di­tion­al hy­po­theses (parts of §6, §7 and §8). In par­tic­u­lar the whole idea of “dual bound­ary” is dis­pensed with here, though this is not to say it should be aban­doned forever. Ref­er­ence to [Chung 1963] bey­ond §4 will be pin­pointed.

Doob’s ver­sion of the fun­da­ment­al con­ver­gence the­or­em of po­ten­tial the­ory as­serts that if \( (f_n) \) is a de­creas­ing se­quence of ex­cess­ive func­tion and \( f \) is the su­per­me­di­an func­tion \( \inf_n f_n \), then the set where \( f \) dif­fers from \( \hat{f} \) (its reg­u­lar­ized func­tion) is semi-po­lar. Many beau­ti­ful proofs of this res­ult are avail­able in the lit­er­at­ure. Here is a trivi­al one.

The Pois­son pro­cess was one of Rényi’s fa­vor­ite top­ics. Here its fa­mil­i­ar prop­er­ties are dis­cussed from the gen­er­al stand­point of re­new­al the­ory, lead­ing to cer­tain simple char­ac­ter­iz­a­tions. Some of the ob­ser­va­tions made be­low are ap­par­ently new des­pite the enorm­ous lit­er­at­ure on the Pois­son pro­cess, oth­ers are stated for pedgo­gic reas­ons–a con­sid­er­a­tion that had al­ways con­cerned Rényi too.

The the­or­ems by Kh­intchine, Koro­ly­uk, and Dobrush­in in the the­ory of sta­tion­ary point pro­cesses are ba­sic and simple the­or­ems. Koro­ly­uk’s the­or­em was ori­gin­ally de­rived from the Palm–Kh­intchine for­mu­las; a dir­ect proof was giv­en in Cramér–Lead­bet­ter [1967]. Its real sim­pli­city seems to be ob­scured by the slightly com­plic­ated present­a­tion of the proof. The same may be said of the proof of Dobrush­in’s the­or­em in­volving an un­ne­ces­sary con­tra­pos­i­tion as well as some ep­si­lon­ics. Both res­ults be­come quiet trans­par­ent when dealt with by stand­ard meth­ods of meas­ure and in­teg­ra­tion in sample space. After all, these are prob­lems of prob­ab­il­ity the­ory and nowadays stu­dents spend a lot of time learn­ing this kind of “ab­stract” set-up. It would be a pity not to use the know­ledge so ac­quired in straight­for­ward situ­ations such as these the­or­ems. In do­ing so we ar­rive at cer­tain nat­ur­al ex­ten­sions which seem to put the res­ults in prop­er per­spect­ive. The res­ults in \( R^d \), ob­tained by the same meth­od, seem to be new.

A com­plete form of the clas­sic­al the­or­em by Gauss–M. Riesz–Frost­man is giv­en for a large of Markov pro­cesses without the usu­al hy­po­thes­is of du­al­ity. The idea leads to a prob­ab­il­ist­ic solu­tion of Robin’s prob­lem and it is based on the last exit time from a tran­si­ent set.

The joint dis­tri­bu­tion of the time since last exit, and the time un­til next en­trance, in­to a unique bound­ary point is giv­en in the fol­low­ing for­mula:
\[ P\{\gamma(t)\in ds:\ \beta(t)\in du\} = E(ds)\,\theta(u-s)\,du \]
for \( s < t < u \). The bound­ary point may be re­placed by a re­gen­er­at­ive phe­nomen­on.

We in­vest­ig­ate the nature of the set of empti­ness times of a dam whose re­lease rate de­pends on the con­tent and whose cu­mu­lat­ive in­put pro­cess is a pure-jump Lévy pro­cess. De­tailed res­ults are ob­tained for stable in­put pro­cesses and re­lease func­tions of the form \( r(x) = x^{\beta}I_{(0,\infty)}(x) \)

The ap­prox­im­a­tion of a giv­en con­tinu­ous para­met­er Markov chain by its dis­crete skel­et­ons raises many in­ter­est­ing ques­tions. Some be­gin­nings of this study were giv­en in §11.13 of [Chung 1967], but little pro­gress seems to have been made since. It is pos­sible that the ap­par­ent im­pen­et­rab­il­ity of the “em­bed­ding prob­lem” (when is a dis­crete para­met­er chain a skel­et­on?) may have un­war­ran­tedly hampered reas­on­able and use­ful in­vest­ig­a­tions. The present note deals with a re­l­at­ively easy as­pect of the ap­prox­im­a­tion the­ory: the glob­al faith­ful­ness of the skelet­al ap­proach as re­flec­ted in the mo­ments of the en­trance time dis­tri­bu­tions. Even so our meth­od leans heav­ily on re­cur­rence. When this hy­po­thes­is is dropped these dis­tri­bu­tions may be im­prop­er, but mo­ments can still be defined by ex­clud­ing the mass at in­fin­ity as done in §1.11 of [Chung 1967] in the dis­crete para­met­er case. Solid­ar­ity res­ults such as The­or­em 1 there can be gen­er­al­ized to the con­tinu­ous para­met­er case without the in­ter­ven­tion of skel­et­ons (see the re­marks con­cern­ing the last part of the the­or­em be­low); but ap­prox­im­a­tion seems an open prob­lem.

Let \( \{W(t):t\geq 0\} \) be the stand­ard Browni­an mo­tion with all paths con­tinu­ous. Let \( M(t)= \max_{0\leq s\leq t}W(s) \) be the max­im­um pro­cess and \( Y(t)=M(t)-W(t) \) be re­flect­ing Browni­an mo­tion. If \( d_{\varepsilon}(t) \) is the num­ber of times \( Y \) crosses down from \( \varepsilon \) to 0 be­fore time \( t \), then it was Paul Lévy’s idea that
\begin{equation*}\tag{1}
P\bigl\{\lim_{\varepsilon\to 0} \varepsilon d_{\varepsilon}(t) = M(t),\
\forall t\geq 0\bigr\} = 1.
\end{equation*}
In [1974] Itô and McK­ean demon­strated the al­most sure con­ver­gence of \( \varepsilon d_{\varepsilon}(t) \) us­ing mar­tin­gale meth­ods. To identi­fy the lim­it they used the hard fact, due to
Lévy, that
\begin{equation*}\tag{2}
P\Bigl\{\lim_{\varepsilon\to 0}
\frac{ \text{measure} \{s: Y(s) < \varepsilon,\ s\leq t\} }{2\varepsilon}=M(t),
\ \forall t\geq 0\Bigr\} = 1
\end{equation*}
and com­puted the second mo­ment of the dif­fer­ence of the ex­pres­sions in (1) and (2). In this pa­per, by ex­amin­ing the ex­cur­sions in Browni­an mo­tion and us­ing a new for­mula for the dis­tri­bu­tion of their max­ima, we ob­tain a dir­ect
iden­ti­fic­a­tion of the lim­it in (1) without us­ing (2).

The con­dens­er the­or­em in clas­sic­al po­ten­tial the­ory is stud­ied with­in the frame­work of Markov pro­cesses and prob­ab­il­ist­ic po­ten­tial the­ory. The con­dens­er charge is ex­pressed in terms of suc­cess­ive balay­ages of a ca­pa­cit­ary meas­ure.

The pur­pose of this pa­per is to for­mu­late and prove sev­er­al ba­sic res­ults for a left con­tinu­ous mod­er­ate Markov pro­cess, which are ana­logues of well-known res­ults for right con­tinu­ous strong Markov pro­cesses.

In this pa­per it is shown that the equi­lib­ri­um meas­ure \( \nu \) for a com­pact \( K \) in po­ten­tial the­ory can be re­lated with a unique in­vari­ant meas­ure \( \pi \) for a dis­crete time Markov pro­cess by the for­mula
\[ \pi(dy) = \phi(y)\,\nu(dy) .\]
The chain has the trans­ition func­tion \( L(x,A) \), where \( L \) is the last-exit ker­nel in [Chung 1973]. For a gen­er­al non-sym­met­ric po­ten­tial dens­ity \( u \) the mod­i­fied en­ergy
\[ I(\lambda) = \iint \lambda(dx)\,u(x,y)\,\phi(y)^{-1}(dy) \]
and the Gauss quad­rat­ic
\[ G(\lambda) = I(\lambda) - 2\lambda(K) \]
are in­tro­duced. Then \( G \) is min­im­ized by \( \pi \) among all signed meas­ures \( \lambda \) on \( K \) of fi­nite mod­i­fied en­ergy, provided \( I \) is pos­it­ive. This in­cludes the clas­sic­al sym­met­ric case of New­to­ni­an and M. Riesz po­ten­tials as a spe­cial case. The modi­fic­a­tion cor­res­ponds to a time change for the un­der­ly­ing Markov pro­cess. The pos­it­iv­ity of \( I \) is es­tab­lished for a class of signed meas­ures as­so­ci­ated with con­tinu­ous ad­dit­ive func­tion­als in the sense of Re­vuz.

The pur­pose of this note is to es­tab­lish a suf­fi­cient con­di­tion for re­cur­rence of a ran­dom walk \( (S_n) \) in \( R^2 \). It fol­lows from it that if \( S_n/n^{1/2} \) is asymp­tot­ic­ally nor­mal then we have re­cur­rence.

In Hunt’s the­ory of Markov pro­cesses cer­tain du­al­ity as­sump­tions are made to gen­er­al­ize well known clas­sic­al po­ten­tial the­or­et­ic res­ults such as F. Riesz rep­res­ent­a­tion the­or­em, unique­ness, ex­ist­ence of equi­lib­ri­um po­ten­tial, etc. A stand­ard treat­ment de­veloped by sev­er­al sub­sequent au­thors can be found in [Blu­menth­al and Getoor 1968]. In a dif­fer­ent dir­ec­tion, it was shown in [Chung 1973] un­der simple ana­lyt­ic con­di­tions that the equi­lib­ri­um meas­ure is in­her­ently linked to the last exit dis­tri­bu­tion of the pro­cess. It thus ap­pears feas­ible that this last res­ult, namely on “equi­lib­ri­um prin­ciple” may be made the start­ing point to which oth­er ma­jor res­ults are re­lated.

In this pa­per we ex­ploit the line of thought in [Chung 1973] to de­rive some of these ma­jor res­ults un­der the same set of con­di­tions as in [Chung 1973]. It turns out that these sets of con­di­tions auto­mat­ic­ally im­ply the ex­ist­ence of a dual. However, this will not be proved here.

Is math­em­at­ics use­ful? We who are en­gaged in the pro­fes­sion must have had oc­ca­sions to won­der about this ques­tion. It is of­ten said that math­em­at­ics is not really a sci­ence; it is an art. It is an art in the sense that in its pur­suit we strive for beauty, not util­ity. If we study math­em­at­ics as an art, per­haps we can jus­ti­fy it for its own sake, al­though there are some people who would dis­par­age “art for art’s sake”. When it comes to sci­ence, there is more reas­on to ask wheth­er it is use­ful. Does it ap­ply to the prac­tic­al needs of our daily life? Does it con­trib­ute to the gen­er­al well-be­ing of so­ci­ety and man­kind?

Des­pite the com­mon use of the term “Feller prop­erty”, there are vari­ations in its defin­i­tion. In the early lit­er­at­ure on Markov pro­cesses, there are dis­cus­sions of this and re­lated prop­er­ties, of­ten un­der sets of be­wil­der­ing as­sump­tions. The coast should now be clear, but cer­tain neat for­mu­la­tions may have been over­looked. In Sec­tion 1 of this note, some old res­ults are re­viewed in more gen­er­al forms and an ap­par­ently new one is de­rived. In Sec­tion 2, the res­ults are ex­ten­ded to in­clude a mul­ti­plic­at­ive func­tion­al, of which prime ex­ample is that of Feyn­man–Kac, prop­erly gen­er­al­ized.

We gen­er­al­ize our pre­vi­ous work on the gauge the­or­em and its vari­ous con­sequences and com­ple­ments, ini­ti­ated in [Chung and Rao 1981] and some­what ex­ten­ded by sub­sequent in­vest­ig­a­tions (see [Chung and Zhao]). The gen­er­al­iz­a­tion here is two-fold. First, in­stead of the Browni­an mo­tion, the un­der­ly­ing pro­cess is now a fairly broad class of Markov pro­cesses, not ne­ces­sar­ily hav­ing con­tinu­ous paths. Second, in­stead of the Feyn­man–Kac func­tion­al, the ex­po­nen­tial of a gen­er­al class of ad­dit­ive func­tion­als is treated. The case of Schrödinger op­er­at­or \( \Delta/2+\nu \), where \( \nu \) is a suit­able meas­ure, is a simple spe­cial case. The most gen­er­al op­er­at­or, not ne­ces­sar­ily a dif­fer­en­tial one, which may arise from our po­ten­tial equa­tions is briefly dis­cussed to­ward the end of the pa­per. Con­crete in­stances of ap­plic­a­tions in this case should be of great in­terest.

In Sec­tion 1, a tem­por­al bound is es­tim­ated by a spa­tial bound, for a Markov pro­cess whose trans­ition dens­ity sat­is­fies a simple con­di­tion. This in­cludes the Browni­an mo­tion, for which com­par­is­on with a more spe­cial meth­od is made. In Sec­tion 2, the res­ult is re­lated to the Green op­er­at­or and ex­amples are giv­en. In Sec­tion 3, the res­ult is ap­plied to an old prob­lem of ei­gen­val­ues of the Lapla­cian. In Sec­tion 4, re­cent ex­ten­sions from the Lapla­cian to the Schrödinger circle-of-ideas are briefly de­scribed. In this case, time is meas­ured by an ex­po­nen­tial func­tion­al of the pro­cess, com­monly known un­der the names Feyn­man–Kac.

This art­icle was writ­ten as script for two talks, one gen­er­al and one tech­nic­al, giv­en in Taiwan in Janu­ary 1991. In the spir­it of the oc­ca­sion, the style of ex­pos­i­tion is de­lib­er­ately paced and dis­curs­ive.

Doeblin re­garded his pa­per [1940] as his hard­est work. The big lim­it is that of \( P^{(n)}(x, E) \) as \( n \) tends to in­fin­ity, in a meas­ur­able non-to­po­lo­gized space. An ex­pos­i­tion of part one of this pa­per was pub­lished in [Chung 1964]. This is the ex­pos­i­tion of part two, which con­tains some re­par­a­tion as well as cla­ri­fic­a­tion.

We con­sider, on the group of in­tegers, a ran­dom walk start­ing from the ori­gin and whose steps ad­mit as pos­sible val­ues ex­actly two in­tegers, \( a \) and \( b \), with \( a < 0 < b \). In the par­tic­u­lar case \( a=-1 \), we give an ex­pli­cit ex­pres­sion for the law of the first re­turn time to the ori­gin.

[176]K. L. Chung:
Green, Brown, and prob­ab­il­ity & Browni­an mo­tion on the line,
2nd edition.
World Sci­entif­ic (River Edge, NJ),
2002.
Part 1 of this book con­sti­tutes the second edi­tion of “Green, Brown and prob­ab­il­ity.” Part 2 com­prises lec­ture notes from the short course on “Browni­an mo­tion on the line,” giv­en at Stan­ford Uni­versity.MR1904742Zbl1014.​60001book

About 1923, the great math­em­atician Paul Lévy in­ven­ted a fam­ily of prob­ab­il­ity dis­tri­bu­tions (laws) called “stable”. If \( X \) and \( Y \) are in­de­pend­ent ran­don vari­ables with the law \( L \), then for any con­stants \( a > 0 \) and \( b \), there ex­ist con­stants \( c > 0 \) and \( d \) such that the law of \( aX + bY \) is the same as \( cZ + d \), where \( Z \) is a ran­don vari­able with the same law \( L \).

This pa­per ex­am­ines — by means of the ex­ample of the St. Peters­burg para­dox — how Borel ex­pos­ited the sci­ence of his day. The first part sketches the sin­gu­lar place of pop­ular­iz­a­tion in Borel’s work. The two parts that fol­low give a chro­no­lo­gic­al present­a­tion of Borel’s con­tri­bu­tions to the St. Peters­burg para­dox, con­tri­bu­tions that evolved over a peri­od of more than fifty years. These show how Borel at­tacked the prob­lem by po­s­i­tion­ing it in a long — and sci­en­tific­ally very rich — med­it­a­tion on the para­dox of mar­tin­gales, those sys­tems of play that pur­port to make a gam­bler toss­ing a coin rich. Borel gave an ori­gin­al solu­tion to this prob­lem, an­ti­cip­at­ing the fun­da­ment­al equal­ity of the nas­cent math­em­at­ic­al the­ory of mar­tin­gales. The para­dox­ic­al role played by Félix Le Dantec in the de­vel­op­ment of Borel’s thought on these themes is high­lighted. An ap­pendix re­casts Borel’s mar­tin­gales in mod­ern terms.

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